Fluid Dynamics and the Navier-Stokes Equations

Contents

Navier-Stokes Equations

Special Cases:


• Incompressible fluid - In fluid dynamics, an incompressible fluid is a fluid whose density is constant. It is the same throughout space and it does not change through time. According to the continuity equation, it also implies u = 0. It is an idealization used to simplify analysis. In reality, all fluids are compressible to some extent.
• Inviscid or Stokes flow - Viscous problems are those in which fluid friction have significant effects on the solution. Problems for which friction can safely be neglected are called inviscid. The Reynolds number (R=(u_sL)/, where u_s is the mean fluid velocity, and L is the characteristic length, e.g., the cross-section of the pipe) can be used to evaluate whether viscous or inviscid equations are appropriate to the problem. High Reynolds numbers indicate that the inertial forces are more significant than the viscous forces. However, even in high Reynolds number regimes certain problems require that viscosity be included. In particular, problems calculating net forces on bodies (such as the wings on aircraft) should use viscous equations. Stokes flow occurs at very low Reynold's numbers, such that inertial forces can be neglected compared to viscous forces.
• Steady flow - Another simplification of the equations is to set all changes of fluid properties with time to zero. This is called steady flow, and is applicable to a large class of problems, such as lift and drag on a wing or flow through a pipe.
• Boussinesq approximation - In fluid dynamics, the Boussinesq approximation is used in the field of buoyancy-driven flow. It states that density differences are sufficiently small to be neglected, except where they appear in terms multiplied by g, the acceleration due to gravity. The essence of the Boussinesq approximation is that the difference in inertia is negligible but gravity is sufficiently strong to make the specific weight appreciably different between the two fluids. Boussinesq flows are common in nature (such as atmospheric fronts, oceanic circulation, downhill winds), industry (dense gas dispersion, fume cupboard ventilation), and the built environment (natural ventilation, central heating). The approximation is extremely accurate for many such flows, and makes the mathematics and physics simpler.
• Laminar vs turbulent flow - Turbulence is flow dominated by recirculation, eddies, and apparent randomness (see Figure 01). Flow in which turbulence is not exhibited is called laminar (see Figure 02). It is believed that turbulent flows obey the Navier-Stokes equations. However, the flow is so complex that it is not possible to solve turbulent problems from first principles with the computational tools available today or likely to be available in the near future. Turbulence is

  		instead modeled using one of a number of turbulence models and coupled with a flow solver that assumes laminar flow outside a turbulent region. Turbulence usually occurs below a Reynold's numbers of 3000. It causes increased energy loss (as heat), more drag (on the moving body), and generates sound wave (noise).
  Figure 01 Turbulent Flow [view large image]	Figure 02 Laminar Flow [view large image]	Modern vehicle and aircraft designs always try to minimize the turbulence by adopting a smooth surface and streamlined contour.

Steady Flow

	0 = - dp / dz + (1/r) d (r d v / dr) / dr ---------- (4) where v denotes the velocity field in the z direction as shown in Figure 02. For a constant pressure drop (per unit length) d p / dz is a negative constant, the solution becomes: v (r) = vm ( 1 - r2 / R2 ) ---------- (5) where vm = (R2 / 4 ) (- dp / dz) ---------- (6)
Figure 03a Cylindrical Co- ordinates [view large image]	is the velocity at the centre, and R is the radius of the pipe. The boundary conditions are v = vm at r = 0 and v = 0 at r = R.

Some comments on steady flow in a pipe:


• Eq.(5) shows that the constant velocity contours are arranged in layers of concentric cylinders with the maximum velocity v_m at the center. There is no mixing of layers. This is one way to define laminar flow.

	The volume flow rate Q is defined as the volume flowing through per unit time: Q = v 2r dr = [(R4)/(8)] (-dp/dz) ---------- (6a) The mass flow rate is just Q x . The flow rate formula in Eq.(6a) is also applicable approximately to the blood flow in the artery. The p and v become temporally varying pulses (Figure 03b). The formula shows that shrinking artery as well as thick blood will
Figure 03b Blood Pressure [view large image]	lower the blood flow rate.

• The definition of v_m in Eq.(6) shows that the flow is driven by the pressure gradient ( - dp / dz). Viscosity is one of the factors in determining the velocity of the flow, which becomes slower for more viscous fluid. The radius of the pipe R also has an influence on the flow, which becomes slower in smaller pipe. The flow comes to a halt with v_m = 0 at the limit of either ( - dp / dz) = 0 or ~ infinity.
• The flow comes to a stop at the wall. This is the consequence of an assumption that the frictional force between the fluid and the wall is infinite. The effect is similar to the thin film of liquid sticking to a solid surface. If this is not the case, then v_m would take the more general form:
  
  v_m = (R² / 4 ) (- dp / dz) + v₀ ---------- (7)
  
  where v₀ is the velocity of the flow at the wall. The solution for the velocity field in Eq.(5) becomes:
  
  v (r) = v_m ( 1 - r² / R² ) + (r / R)² v₀---------- (8)

In case when the pressure gradient is absent, i.e., (- dp / dz) = 0 in Eqs.(4) and (7). The above solution is reduced to v(r) = vm = v0, that is, the flow has an uniform velocity similar to the case shown in Figure 04a for the uniform flow of inviscid fluid. But in this example, the viscosity term disappears in Eqs.(4) and (7) because (- dp /dz) = 0; and thus has no influence on the flow. It seems that the flow can last forever without a driving force. In reality, it would finally come to a halt by the dissipative effects and the discharge at the end of the pipe (in this example an infinitely long pipe is assumed).

Figure 04a Uniform Flow [view large image]

	Instead of maintaining the pressure gradient by a pump, the flow can be driven by gravity if it is tilted by an angle A with respect to the horizon (Figure 04b). Then (- dp / dz) = g sin(A) ---------- (9) where is the density of the fluid, g = 980 cm/sec2 is the acceleration of gravity, and
Figure 04b Flow by Gravity Feed [view large image]	sin(A) is related to the slant of the pipe. Such method is a convenient way to transport liquid, but it does not work for gas since its density is about 1000 times lower.

Lift for Aeroplane and Helicopter

	where the subscripts 1 and 2 respectively denote the flow velocity ux, and the pressure p to the upper and lower layers of the air flow. It shows that if the structure of the wings are designed to create a higher flow velocity in the upper layer, then a net pressure in the
Figure 05 Lift for Aeroplane [view large image]	upward direction is created to lift the aeroplane up into the air provided such force can overcome the weight [see Figure 05, the length of the arrows is proportional to the magnitude of ux (blue) and p (red)]. The force to lift up the plane is just f = P x Sref ,

	where P = p2 - p1 is the net upward pressure, and Sref (Reference Area) is the effective area acted upon by P. Figure 06 shows the Sref for an aircraft and a helicopter. The actual force F on these objects is determined further by the formula F = f x CL, where CL is the Coefficient of Lift. It depends on the type of aircraft as shown in Figure 06, where the angle of attack is
Figure 06 Ref. Area & Lift Coeff. [view large image]	defined as the angle between the wing and the direction of the airstream. Most aircrafts will behave similarly to the Cessna 172 while high-speed planes with short wingspans, like fighters, will more closely resemble the Lightning data.

	The lift for helicopter can be derived from Eq.(2) of the Navier-Stokes Equations in a similar way (as for the aeroplane) with the role of ux and uz interchanged because the airflow pattern is different (see Figure 07). Thus for uz >> ux , Eq.(10) becomes: uz [d(uz)/dz] = - (1/) dp/dz ---------- (12) Integrating Eq.(12) yields: { [(-uz2)2 - (-uz1)2]} / 2 = (p2 - p1) ---------- (13)
Figure 07 Helicopter Airflow [view large image]	where uz is negative as it is pointing toward the negative z direction. It has a form similar to Eq.(11) for the aeroplane. Computation for the lifting force follows exactly the same line as developed previously.

	In stationary position, the helicopter's engine provides only the lifting force. According to the principle of angular momentum conservation, the body would turn in the opposite direction of the rotating blades. To stabilize the helicopter, a tail rotate is installed to counteract this trend. By applying more or less pitch angle to the tail rotor blades, it can
Figure 08 Helicopter [view large image]	also be used to make the helicopter turn left or right. Forward motion is achieved by tilting the spinning rotor in the direction of the flight (see Figure 08). There are many factors related to the rotor blades to limit the maximum speed of a helicopter at about 400 km/h.

By rotating the rotor blades and the z-axis 90o, Eq. (13) is applicable to calculate the forward pressure (or thrust) of a propeller on a ship or airplane (Figure 09). Since the density for water is about 1000 times higher than air, the formula shows that the propeller can generate more thrust with lower flow speed on a ship or boat. It also explains why the prop plane cannot fly high up into the rare atmosphere, where the air density is very low.

Figure 09 A Prop Plane [view large image]

High and Low Pressure Cells

		relatively high, central-pressure zone. As the air diverges from the central region, it is deflected by the Coriolis force in a clockwise circulation (Figures 10 & 12). Thus, most Highs are generally elliptical in shape following their formation. But as they interact with other air masses and topography, and are distorted by forces of the upper atmosphere, high pressure cells often become long and
Figure 10 High Pressure Ridge	Figure 11 Low Pressure Cell [view large image]	narrow in shape, and is referred to as high pressure ridge in the weather map. Since the air at high altitude is dry, the High is usually associated with fair weather.

Mathematically, the sinking and rising air can be explained by Archimedes' principle as discussed in the section of Hydrostatics. Cooler air mass will sink as it is denser than the surrounding air, and vice versa for the warmer air mass. The swirling motion of air on the horizontal plane is determined by the Navier-Stokes Equations in Eq.(2). Since the radial velocity ur is usually much smaller than the circular velocity v = r u in the core region, the radial component of the equation (in cylindrical coordinates) can be reduced to:

Figure 12 Coriolis Force [view large image]

2 v sin = (1/)p/r ---------- (14a)

The Great Red Spot (Figure 13) in Jupiter provides a very good example to illustrate the Coriolis force at work. It is a high pressure cell located 22o South of the equator. Thus, the rotational vector is pointing inward to the center (instead of pointing outward as for the case in the Northern hemisphere), and the swirling gas is circulating counter-clock wise. This system of anticyclonic storm has existed for up to 400 years. The long lifetime cannot be attributed entirely to the higher rotational speed (about twice as much as that for the Earth), and hence the stronger Coriolis force. It is suggested that the lack of solid surface to

Figure 13 Great Red Spot [view large image]

provide friction may play a part contrary to the hurricane on Earth, which always break up shortly after landfall. Note that the oval shape is caused by the constriction from the neighboring cloud bands.

Spiral Flow

		phenomena such as spiral galaxy, hurricane, and drain in the sink (Figures 14, and 15). By neglecting the thickness of the spiral flow, Eq.(2) of the Navier-Stokes Equations in cylindrical coordinates can be expressed in the form: ur (ur/r) + u (ur/) = Fr ---------- (15a) ur [(r u)/r] + u [(r u)/] = 0 ---------- (15b) where we have assumed further that the force acts on the fluid
Figure 14 Barred Spiral Galaxy [view large image]	Figure 15 Hurricane [view large image]	only in the radial direction. In addition by assuming constant density for the fluid, Eq.(1) of the continuity equation becomes: (1/r) [(r ur)/r] + (1/r) [(r u)/] = 0 ---------- (16)

	A relationship between the velocity components is obtained by substituting Eq.(16) to Eq.(15b): ur = b u ---------- (17) where b is a constant having the dimension of length. This formula can be integrated once more to yield: r = a + b ---------- (18)
Figure 16 Archimedean Spiral [view large image]	where a is another constants of integration. Eq.(18) expresses the trajectory of an Archimedean spiral (see curve on the left of Figure 16). So far there is no restriction on whether the movement is to plunge inward or to expand.

The appearance of the spiral is determined by the constants of integration:
1. The value of b determines the winding of the spiral. As b = dr/d is the relative rate of change between r and , a small value of b makes the winding very tight and vice versa.
2. The solution also admits another spiral arm with - or 180^o out of phase (see curves on the right of Figure 16).
3. With variation of orientation at r 0, the spiral assumes a broad sweeping pattern much like the hurricane in Figure 15 instead of one slim locus.
4. The sign of b, i.e., b > 0 or b < 0, has the effect on the winding direction - whether it turns to the left or right.
5. For the barred spiral like the Milkyway in Figure 14, the constant a > 0, while a = 0 is for the case of true spiral galaxy such as NGC2997. It turns out that the formation of spiral arms in galaxy is more complicated than this simple minded approach, which would produce tightly wound spirals (within 500 million years) in contrary to observation (see more in Theory of Spiral Arm Formation).

with M to be the mass of the central black hole, while the mass on the disk within r is given by M' = 2 r dr, where = 0 (r0/r) is the surface density in unit of gm/cm2, 0 is its cutoff value at the edge of the galaxy r = r0, and the third term is the centrifugal force. Thus, the radial and rotational components of the velocity can be expressed as: ur2 = GM / r + 2G0r0 - r2u2 ---------- (19b), r2u2 = r (GM + 2G 0r0 r) / (b2 + r2) ---------- (19c),

Figure 17a Rotation Curve for Milky Way [view large image]

which has the same profile of the rotation curve for the disk as shown in Figure 17a with a peak at rm (40r0b/M) b for 40r0b >> M. The observed curve takes into account of the dark matter in the halo.

dp/dr (which has overwhelmed the Coriolis and centrifugal forces in this case): ur2 / r = (1/) dp/dr ---------- (20). The dependence of p on r is shown in Figure 17b. The observational data (for the "Charlie" type hurricane) can be approximated by the empirical formula: p = p0 [5.5 - e-k(r - re)] ---------- (21) for r re, where re is the distance from the center to the wall of the eye (Figure 17b), p0 = 220 mb and k is a constant related to the steepness of the pressure gradient. Thus, the pressure gradient is: dp/dr = kp0e-k(r - re) ---------- (22) for r re or ur2 = (krp0/) e-k(r - re) ---------- (23) for r re. At the wall of the eye, r = re, and ur2 = krep0/.

Figure 17b Pressure Gradient of Hurricane

The rotation curve is given by r2u2 = (kr3p0/b2) e-k(r - re), which reproduces a profile similar to the curve in Figure 17b with a maximum at rm = 3/k. This kind of analysis is also applicable to the drain in the sink although on a much smaller scale.

Hydrostatics

		The term "hydrostatics" is applied to the study of fluids at rest, including both liquids and gases. For this special case, u = 0 in the Navier-Stokes equations, the external force F = -g (in the negative y direction) and the pressure gradient is just dp. Eq.(2) is reduced to the simple form: dp = - g dy ---------- (24). Taking point 1 and 2 as shown in Figure 18, integration of Eq.(24) yields: p2 - p1 = - g (y2 - y1) ---------- (25)
Figure 18 Hydrostatics [view normal image]	Figure 19 Atmosphere [view large image]	which shows that the pressure is less in higher elevation as it is well known in the Earth's atmosphere (Figure 19).

Submarine controls the submerging and surfacing by admitting water into a tank or pumping it out. When the submarine is on the surface its ballast tanks are full of air (Figure 20). The air inside the hull brings the average density of the entire ship below the density of saltwater. Because the sub is less dense, it floats. The average density of a submerging submarine is increased by flooding the ballast tanks with water through flood ports on the bottom of the tanks with the release of air out of the top of the tanks (Figure 20). Once the overall density of the submarine is equal to the water around it, it has the neutral buoyancy and will remain at that depth. High pressure air is blown into the ballast tanks to push the water out when the submarine is prepared to surface again.

Figure 20 Submarine [view large image]

Air does not provide a lot of buoyant force because its density is about 1000 times lower. Only balloon filled with low density gas such as hydrogen or helium can rise up to the sky.

Formation of Spherical Body

		Solid body does not flow like fluid, the Navier-Stokes Equations doesn't seem to be applicable in such case. However, the equation can be used to calculate the critical mass for the self-gravity of a solid body to overcome its resistant forces so that it assumes a hydrostatic equilibrium (nearly round) shape. Solid
Figure 21 Deformation of Solid Body	Figure 22 Geometric Shape [view large image]	body actually has elastic property, that makes it to deform in response to the external force as shown by the deformation of the moon by the gravitational tug of a planet in Figure 21.

Table 01 Geometric Shape of Solid Body